L(s) = 1 | + (0.783 + 0.621i)2-s + (−0.239 + 0.970i)3-s + (0.227 + 0.973i)4-s + (0.275 + 0.961i)5-s + (−0.791 + 0.611i)6-s + (−0.996 − 0.0868i)7-s + (−0.426 + 0.904i)8-s + (−0.885 − 0.465i)9-s + (−0.381 + 0.924i)10-s + (−0.215 + 0.976i)11-s + (−0.999 − 0.0124i)12-s + (0.948 + 0.317i)13-s + (−0.726 − 0.687i)14-s + (−0.999 + 0.0372i)15-s + (−0.896 + 0.443i)16-s + (0.969 − 0.245i)17-s + ⋯ |
L(s) = 1 | + (0.783 + 0.621i)2-s + (−0.239 + 0.970i)3-s + (0.227 + 0.973i)4-s + (0.275 + 0.961i)5-s + (−0.791 + 0.611i)6-s + (−0.996 − 0.0868i)7-s + (−0.426 + 0.904i)8-s + (−0.885 − 0.465i)9-s + (−0.381 + 0.924i)10-s + (−0.215 + 0.976i)11-s + (−0.999 − 0.0124i)12-s + (0.948 + 0.317i)13-s + (−0.726 − 0.687i)14-s + (−0.999 + 0.0372i)15-s + (−0.896 + 0.443i)16-s + (0.969 − 0.245i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4445424616 + 1.459872335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4445424616 + 1.459872335i\) |
\(L(1)\) |
\(\approx\) |
\(0.6588679443 + 1.125993508i\) |
\(L(1)\) |
\(\approx\) |
\(0.6588679443 + 1.125993508i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.783 + 0.621i)T \) |
| 3 | \( 1 + (-0.239 + 0.970i)T \) |
| 5 | \( 1 + (0.275 + 0.961i)T \) |
| 7 | \( 1 + (-0.996 - 0.0868i)T \) |
| 11 | \( 1 + (-0.215 + 0.976i)T \) |
| 13 | \( 1 + (0.948 + 0.317i)T \) |
| 17 | \( 1 + (0.969 - 0.245i)T \) |
| 19 | \( 1 + (-0.926 - 0.375i)T \) |
| 29 | \( 1 + (0.481 - 0.876i)T \) |
| 31 | \( 1 + (0.606 + 0.794i)T \) |
| 37 | \( 1 + (-0.514 - 0.857i)T \) |
| 41 | \( 1 + (0.524 + 0.851i)T \) |
| 43 | \( 1 + (-0.952 - 0.305i)T \) |
| 47 | \( 1 + (0.203 + 0.979i)T \) |
| 53 | \( 1 + (-0.381 - 0.924i)T \) |
| 59 | \( 1 + (-0.982 - 0.185i)T \) |
| 61 | \( 1 + (0.545 - 0.837i)T \) |
| 67 | \( 1 + (0.346 + 0.938i)T \) |
| 71 | \( 1 + (0.992 + 0.123i)T \) |
| 73 | \( 1 + (0.481 + 0.876i)T \) |
| 79 | \( 1 + (0.827 + 0.561i)T \) |
| 83 | \( 1 + (0.00620 - 0.999i)T \) |
| 89 | \( 1 + (0.369 - 0.929i)T \) |
| 97 | \( 1 + (-0.263 + 0.964i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−23.2224143899383900052593655736, −22.18169803403466125624916774055, −21.29144214483252547840270708370, −20.52170416702517253784356697429, −19.5870363381453868402911406841, −18.9713387469029349788841683278, −18.27300514830887317518918312275, −16.86836491425805929347412861752, −16.31704971654576698838474002342, −15.238014719274753464547369439656, −13.748423269024802120768488953106, −13.55719358282584333075277234681, −12.541786839712868963054875574633, −12.22636898538020587057580470354, −11.013112805293942033600902915837, −10.15483331946157166078414661766, −8.92751550153815077865663375550, −8.1036779728171161015785107577, −6.52259915898378650351878974866, −5.9596682299892435771300445715, −5.239315404857244542865078047175, −3.770336661227176111130872847187, −2.85607960731030527057003546470, −1.582373540276344881575942197381, −0.63827787709188252966135999691,
2.47083326794966682857089733997, 3.36397337698642939411567218302, 4.12545191159268821879811345708, 5.24231674966487720433676213424, 6.30283953280845235487091356742, 6.72249481556175766363991443943, 8.02271783449127293769014198629, 9.28149777640530401939427319839, 10.120876749597800008778763242256, 10.97966764458115365944366958516, 11.98626997497434921123664585614, 12.96259930992269670061799206092, 13.97999735625024676789945447125, 14.66606086579888335441639123847, 15.59117997695741348684194044434, 15.98550140029041315570511909680, 17.0644384432506206885808653467, 17.733206621833113105492741069041, 18.882725967326779319427467746382, 20.03949236647178586562168048441, 21.12631458865914234300023965041, 21.501691529026593605425058077594, 22.5266401297503995734813497950, 23.154022540814699859329875284, 23.3226916707246069180924557440